Geophysics lecture chapter 5 geodynamics

Chapter 5


Geodynamics

5.1

Heat flow

Thermally controlled processes within Earth include volcanism, intrusion of
igneous rocks, metamorphism, convection within the mantle and outer core,
and plate tectonics. The global heat flow can be measured by measuring the
temperature gradient everywhere at the surface of the Earth. This gives us an
estimate of the mean rate of heat loss of the Earth, which can be broken up into
various components (Table 7.3 in Fowler):

Continents
Oceans

Area
(km2 )
201
309
Conductive cooling
Hydrothermal circulation

Total Earth

510

Heat Flow

(mWm−2 )
58
100
[66]
[34]
83

Heat Loss
(1012 W)
11.5
30.4
[20.3]
[10.1]
41.9

The amount of heat lost through the ocean basins in enormous! — up to 73%!
(The oceans cover about 60% of the Earth’s surface). This was a famous paradox
before the discovery of plate tectonics. It was well known that the abundance
of radioactive elements (which are a source of heat through radioactive decay)
in the ocean basins was much lower than that in the continents. So what causes
the significantly higher heat flow in the oceans? With the discovery of plate
tectonics it was realized that most of the heat loss occurs through the cooling
and creation of oceanic lithosphere. The mean rate of plate generation therefore
depends on the balance between the rate of heat production within the Earth
and the rate of heat loss at the surface.
In this course we will address some of the basic concepts of heat flow and
Earth’s thermal structure, and we will discuss in some detail the cooling of
oceanic lithosphere and the implications of Earth thermal structure for mantle
convection.
191

192

CHAPTER 5.

GEODYNAMICS

Heat sources
There are several possibilities for the source of heat within the earth:
1. ”Original” or ”primordial” heat; this is the release of heat due to the
cooling of the Earth. The amount of heat released by this process can be
estimated by calculating the heat released by a change in temperature of
1◦ at constant pressure. This depends on the specific heat, CP which
is the energy that is needed to heat up 1 kg of material by 1◦ (i.e., it’s a
material property).
We can do a quick calculation to find out how much heat would be released
by dropping the temperature of the mantle by 1◦ C (Let’s for now ignore
latent heat due to phase changes):
• Mantle; for silicates:CP = 7.1 × 102 Jkg−1◦ C−1 ; the mass of the
mantle is about 4.1×1024 kg
• Core; for iron: CP = 4.6 × 102 Jkg−1◦ C−1 ; the mass of the core is
about 1.9×1024 kg
For ΔT = 1◦ C this gives ΔE = 3.7×1027J. In absence of any other sources
for heat production, the observed global heat flux of 4.2×1013W can thus
be maintained by a cooling rate of 4.2×1013 [W] divided by 3.7×1027 [J]=
1.1×10−14◦ Cs−1 .
In other words, since the formation of Earth, 4.5 Ga ago, the average
temperature would have dropped by ΔT ≈ 1, 500◦C. Note that the actual
cooling rate is much lower because there are sources of heat production.

2. Gravitational potential energy released by the transfer of material from
the surface to depths. Imagine dropping a small volume of rock from the
crust to the core. The gravitational potential energy released would be:
ΔE = Δρgh, with g ≈ 10ms−2 and h = 3 × 106 m
ρsilicates ≈ 3 × 103 kgm−3 and ρiron ≈ 7 × 103 kgm−3 , so that Δρ = 4 × 103
kgm−3 .
ΔE ≈ 1.2 × 1011 Jm−3 .

The present-day heat flux would thus be equivalent to dropping a volume
of about 350 m3 every second. This is equal to dropping a 22 m thick
surface layer every million years.
So even if a small amount of net differentiation were taking place within
the earth, this would be a significant source of heat!
3. Radioactive decay: for an order of magnitude calculation, see Stacey 6.3.1.
The bottom line is that for the Earth a very significant fraction of heat
loss can be attributed to radioactive decay (primarily of Uranium (U),
Thorium (Th) and Potassium (K). More, in fact, than can be accounted
for by heat production of the MORB source.


193

5.1. HEAT FLOW
Heat transfer

The actual cooling rate of the Earth depends not only on these sources of heat,
but also on the efficiency at which heat is transferred to and lost at the Earth’s
surface.
How does heat get out of the system?
Conduction — this will be discussed below in the context of the cooling of

oceanic lithosphere.
Convection — For example, in the mantle and core.
Radiation — most of the heat that the Earth receives from external sources
(i.e. the Sun) is radiated out.
Radiation
The net effect is that the Earth is cooling at a small rate (of the order of
50-100◦C per Ga!) (See Stacey (1993), p. 286.)

Figure 5.1:


194

5.2

CHAPTER 5.

GEODYNAMICS

Heat flow, geothermal gradient, diffusion

The rate of heat flow by conduction across a thin layer depends on
1. the temperature contrast across the layer (ΔT )
2. the thickness of the layer (Δz)
3. the ease with which heat transfer takes place (which is determined by
the thermal conductivity k). The thinner the layer and the larger the
temperature contrast (i.e., the larger the gradient in temperature), the
larger the heat flow.
In other words, the heat flow q at a point is proportional to the temperature
gradient at that point. This is summarized in Fourier’s Law of conduction:

q = −k∇T ≈ −k

ΔT
z
ˆ
Δz

(5.1)

where the minus sign indicates that the direction of heat flow is from high
to low tempertaures (i.e., in the opposite dirtection of z if z is depth.). (For
simplicity we talk here about a 1D flow of heat, but Fourier’s Law is also true
for a general 3D medium).
We can use this definition to formulate the conduction (or diffusion) equation, which basically describes how the temperature per unit volume of material
changes with time. This change depends on
1. the amount of heat that flows in or out of the system which is described
by the divergence of heat flow
2. the amount of heat produced within the volume (denoted by the density
of heat sources A)
3. the coupling between this change in heat and a change in temperature
(which is controlled by the specific heat)
The thermal diffusion equation is given by:
ρCP

∂T
= −∇ · q + A
∂t

(5.2)

Or: the change in heat content with time equals the divergence of the heat
flow (into and out of the volume) and the generation of heat within the volume.
Combined with Fourier’s Law the diffusion equation can be written as
ρCP

∂T
= −∇ · (−k∇T ) + A = k∇2 T + A
∂t

(5.3)

In a situation of steady-state the diffusion equation transforms to the expression of the geotherm, the variation of temperature with depth in the Earth:
k∇2 T + A = ρCP

∂T
A
= 0 ⇒ ∇2 T = −
∂t
k

(5.4)


5.2. HEAT FLOW, GEOTHERMAL GRADIENT, DIFFUSION

195

If there is no heat production (by radioactive decay), i.e., A = 0, then
the temperature increases linearly with increasing depth. If A �= 0 then the
temperature/depth profile is given by a second-order polymomial in z. In other

words, the curvature of the temperature-depth profile depends on the amount
of heat production (and the conductivity).

Figure 5.2: Heat production causes nonlinear geotherms.
A typical value for the geotherm is of order 20 Kkm−1 , and with a value
for the conductivity k = 3.0 Wm−1 K−1 this gives a heat flow per unit area of
about 60 mWm−2 (which is close to the global average, see table above). If the
temperature increases according to this gradient, at a depth of about 60 km a
temperature of about 1500 K is reached, which is close to or higher than the
melting temperature of most rocks. However, we know from the propagation
of shear waves that the Earth’s mantle behaves as a solid on short time scales
(µ > 0). So what is going on here? Actually, there are two things that are
important:
1. At some depth the geothermal gradient is no longer controlled by conductive cooling and adiabatic compression takes over. The temperature
gradient for adiabatic compression (i.e., the change of temperature due
to a change of pressure alone, without exchange of heat with its invironment) is much smaller than the gradient in the conducting thermal
boundary layer.
2. With increasing pressure the temperature required for melting also increases. In fact it can be shown that with increasing depth in Earth’s
mantle, the actual temperature increases (from about 0◦ C at the surface to about 3,500 ± 1000◦ C at the core-mantle boundary CMB) but
the melting temperature Tm increases even more as a result of the increasing pressure. Consequently, at increasing depth in the mantle the
ratio of T over Tm (the homologous temperature) decreases. At even
larger depth, in Earth’s core, the temperature continues to increase, but
the melting temperature for pure iron drops (pure chemical compounds
— such as pure iron — typically have a lower melting temperature then


196

CHAPTER 5.

GEODYNAMICS

most mixtures — such as silicate rock) so that the actual temperaute ex
ceeds the melting temperature and the material is in liquid state. Even
tough the mantle is ’solid’ it behaves as highly viscous fluid so that flowis possible over very long periods of time.

5000

5000

4000
Solidus
3000

3000
Temperature

2000
2000

Temperature (oC)

Temperature (K)

4000

1000
1000
0

0
0

A

2000

D''

6000

4000

Depth (km)

6371

5150

2891

400
670

L

Inner
Core

Outer Core

Lower Mantle

L = Lithosphere (0-80 km)

A = Asthenosphere (80-220 km)

D'' = Lower-Mantle D'' Layer

400, 670 = Phase Transitions

Figure by MIT OCW.

Figure 5.3: Geotherms in the Earth.


If we ignore heat production by radioactive decay we can simplify the con-

duction equation to

ρCP

k
∂T
∂T
∇2 T = κ∇2 T
= k∇2 T ⇒
=
ρCP
∂t

∂t

(5.5)

with κ the thermal diffusivity
κ=

k
ρCP

(5.6)

We will look at solutions of the diffusion equation when we discuss the cooling
of oceanic lithosphere after its formation at the mid oceanic ridge. Before we
do that let’s look at an important aspect of the diffusion equation.


5.3. THERMAL STRUCTURE OF THE OCEANIC LITHOSPHERE 197
From a dimensional analysis of the diffusion equation
∂T
= κ∇2 T
∂t

(5.7)

we see that the diffusivity κ has the dimension
of length2 × time−1 . We can
√
now define a diffusion length L as L = κt.
If a temperature change occurs at some time t0 , then after√a characteristic

time interval τ it will have ’propagated’ over a distance L = κτ through the
medium with diffusivity κ. Similarly, it takes a time l2 /κ for a temperature
change to propagate over a distance l.

5.3

Thermal structure of the oceanic lithosphere

Introduction
The thermal structure of the oceanic lithosphere can be constrained by the
observations of:
1. Heat flow
2. Topography (depth of the ocean basins)
3. Gravity (density depends inversely on temperature)
4. Seismic velocities (µ = µ(T ), λ = λ(T )); in particular, surface waves are
sensitive to radial variations in wave speed and surface wave dispersion
is one of the classical methods to constrain the structure of oceanic (and
continental) lithosphere.
In the following we address how the heat flow and the depth of ocean basins
is related to the cooling of oceanic lithosphere.
The conductive cooling of oceanic lithosphere when it spreads away from the
mid-oceanic ridge can be described by the diffusion equation
∂T
= κ∇2 T + A
∂t

(5.8)

We will simplify this equation by (1) ignoring the heat production by radiocative decay, so that A = 0 (this is reasonable for the oceanic lithosphere
since the basalts do not contain a significant fraction of major radio-isotopes

Uranium, Potassium, and Thorium)1 , and (2) by assuming a 2D geometry so
that we can ignore the variations in the y direction. The latter assumption is
justified for regions away from fracture zones. With these simplifications the
diffusion equation would reduce to a
�
� 2
∂ T
∂2T
∂T
(5.9)
= κ∇2 T = κ
+
∂t
∂x2
∂z 2


198

CHAPTER 5.

GEODYNAMICS

with z the depth below the surface and x the distance from the ridge. The
variation in temperature in a direction perpendicular to the ridge (i.e., in the
spreading direction x) is usually much smaller than the vertical gradient. In that
case, the heat conduction in the x direction can be ignored, and the cooling of
a piece of lithosphere that moves along with the plate, away from the ridge, can
be described by a 1D diffusion equation:
∂T

=κ
∂t

�

∂ 2T
∂z 2

�

(5.10)

(i.e., the ’observer’, or the frame of reference, moves with the plate velocity
u = x/t). Note that, in this formulation, the time plays a dual role: it is used
as the time at which we describe the temperature at some depth z, but this also
relates to the age of the ocean floor, and thus to the distance x = ut from the
ridge axis).

u

Ridge
x=0

SURFACE

u

T = Ts

x

LITHOSPHERE
ISOTHERM

Tm
ASTHENOSPHERE

y
Figure by MIT OCW.

Figure 5.4: The cooling of oceanic lithosphere.

Ridge

T = Ts

u

T = Tm

u
q

q
t=0

x
t= u

u

q

t = t1

t = t2

Figure by MIT OCW.

Figure 5.5: Bathymetry changes with depth.


5.3. THERMAL STRUCTURE OF THE OCEANIC LITHOSPHERE 199
The assumption that the oceanic lithosphere cools by conduction alone is
pretty good, except at small distances from the ridge where hydrothermal circulation (convection!) is significant. We will come back to this when we discuss
heat flow. There is a still ongoing debate as to the success of the simple cooling
model described below for large distances from the ridge (or, equavalently, for
large times since spreading began). This is important since it relates to the
scale of mantle convection; can the cooling oceanic lithosphere be considered as
the Thermal Boundary Layer (across which heat transfer occurs primarily by
conduction) of a large scale convection cell or is small scale convection required
to explain some of the observations discussed below? See the recent Nature
paper by Stein and Stein, Nature 359, 123–129, 1992.

Cooling of oceanic lithosphere: the half-space model
The variation of temperature with time and depth can be obtained from solving
the instant cooling problem: material at a certain temperature Tm (or T0 in
Turcotte and Schubert) is instantly brought to the surface temperature where
it is exposed to surface temperature Ts (see cartoons below; for a full derivation,
see Turcotte and Schubert).

y=0

T0

T

y=0

t = 0­

y

Ts

T0

T

y=0

T0

t = 0+

Ts

T

t>0

y

y

Figure by MIT OCW.

Figure 5.6: The heating of a halfspace
Diffusion, or relaxation to some reference state, is described by error functions2 , and the solution to the 1D diffusion equation (that satisfies the appropriate boundary conditions) is given by

�

z
T (z, t) = Tz (t) = Ts + (Tm − Ts ) erf √
2 κt


�

(5.11)

or

�

z
T (z, t) − Ts = (Tm − Ts ) erf √
2 κt

�

(5.12)

with T (z, t) the temperature within the cooling boundary layer, Ts and Tm
the temperature at the surface and in the mantle, respectively, κ the thermal
2 So

called because they are integrations of the standard normal distribution.


200

CHAPTER 5.

GEODYNAMICS

diffusivity3 , κ = k/ρCp (k is the thermal conductivity and Cp the specific
heat), and the error function operating on some argument η defined as
2
erf(η) = √
π

�η

2

e−u du

(5.13)

0

The so called complementary error function, erfc, is defined simply as erfc(η) =
1 − erf(η). The values of the error function (or its complement) are often presented in table form4 . Figure 5.7 depicts the behavior of the error function:
when the argument increases the function value ’creeps’ asymptotically to a
value erf = 1.

1.0
0.8

erf �

0.6
0.4
erfc �

0.2
0

0

1

2

3

�
Figure by MIT OCW.

Figure 5.7: Error function and complimentary error function.

Let’s look at the temperature according to (5.12) for different boundary
conditions. For large values of z the solution of the diffusion equation becomes
T (∞, t) = Tm ; at the surface, z = 0 so that T (0, t) = Ts , and= after a very long
3 The thermal diffusivity κ has the dimension of distance2 /time; a typical value for κ is 1
√
mm2 /s. The square root of the product κt is porportional to the diffusion length L ∼ κt.
If the temperaure changes occur over a characteristic time interval t they will propagate a
distance of the order of L. Similarly, a time l2 /κ is required for temperature changes to
propagate distance l.
4 Type help erfin MatlabTM


5.3. THERMAL STRUCTURE OF THE OCEANIC LITHOSPHERE 201
time, T (z, ∞) = Ts , i.e., the whole system has cooled so that the temperature
is the same as the surface temperature everywhere.
√
For the Earth we can set Ts = 0◦ C so that T ≈ Tm erf(η), η = z/(2 κt)
for most practical purposes; but the above formulas are readily applicable to
other boundary layer problems (for instance to the cooling of the lithosphere on
Venus where the surface temperature is much than that at Earth).
Examples of the geothermal gradient as a function of lithospheric age are
given in the diagram below (from Davies & Richards, ”Mantle Convection”
J. Geol., 100, 1992).

Temperature (oC)
0

0

1000

5 Myr

Depth (km)

25 Myr
100 Myr

Dry solidus

100
Craton

200

Figure by MIT OCW.

Figure 5.8: Cratonic and oceanic geotherms.

Figure 5.9 (from Turcotte & Schubert, 1982) shows a series of isotherms
(lines of constant temperature (i.e, T (z, t) − Ts = constant) for Tm − Ts =
1300◦C; it shows that the depth to the isotherms as defined by (2) are hyperbola.
From this, one can readily see that if the thermal lithosphere
is bounded by
√
isotherms, the thickness of the lithosphere√increases as t. For back-of-theenvelope calculations you can use D ∼ 2.3 κt for lithospheric thickness. (For
κ = 1 mm2 /s and t = 62.8 Ma, which is the average age of all ocean oceanic


202

CHAPTER 5.

GEODYNAMICS

t (Myr)
0

0

50

100

150
200

y (km)

400
600

50
800
1000

100

150

Figure by MIT OCW.

Figure 5.9: Oceanic geotherms.
lithosphere currently at the Earth’s surface, D ∼ 104 km). This thickening
occurs because the cool lithosphere reduces the temperature of the underlying
material which can then become part of the plate. On the diagram the open
circles depict estimates of lithospheric thickness from surface wave dispersion
data. Note that even though the plate is moving and the resultant geometry
is two dimensional the half space cooling model works for an observer that is
moving along with the plate. Beneath this moving reference point the plate is
getting thicker and thicker.


5.3. THERMAL STRUCTURE OF THE OCEANIC LITHOSPHERE 203
Intermezzo 5.1 lithospheric thickness from surface wave
dispersion
In the seismology classes we have discussed how dispersion curves can be used
to extract information about lithospheric structure from the seismic data. The
thickness of the high wave speed ’lid’, the structure above the mantle low velocity zone, as determined from surface wave dispersion across parts of oceanic
lithosphere of different age appears to plot roughly between the 900◦ and 1100◦ C
isotherms, or at about T = 0.8 − 0.9Tm (see Figure 5.10). So the seismic ”lithosphere” seems to correspond roughly to thermal lithosphere. In other words; in
short time scales — i.e. the time scale appropriate for seismic wave transmission
(sec - min) — most of the thermal lithosphere may act as an elastic medium,
whereas on the longer time scale the stress can be relaxed by steady state creep,
in particular in the bottom half of the plate. However, a word of caution is in order since this interpretation of the dispersion data has been disputed. Anderson
and co-workers argue (see, for instance, Anderson & Regan, GRL, vol. 10, pp.
183-186, 1983) that interpretation of surface wave dispersion assuming isotropic
media results in a significant overestimation of the lid thickness. They have
investigated the effects of seismic anisotropy and claim that the fast isotropic
LID extends to a much cooler isotherm, at T ∼ 450◦ − 600◦ C, than the base of
the thermal lithosphere.

Heat flow
If we know the temperature at the surface we can deduce the heat flow by
calculating the temperature gradient:

q

=
=

∂T
∂T ∂η
= −k
∂z
∂η ∂z
k(Tm − Ts ) ∂
k ∂
√
erf(η)
− √
(Tm − Ts ) erf(η) = −
∂η
∂η
2 κt
2 κt

−k

(5.14)

with

∂
erf(η)
∂η

=

∂ 2
√
∂η π

�η

2 ∂
√
π ∂η

�η

2

e−u du

0

=

2
2

2
e−u du = √ e−η
π

(5.15)

0

so that
q = −k

(Tm − Ts ) −η2
√
e
πκt

(5.16)


204

CHAPTER 5.

0

Age (Ma)

0

GEODYNAMICS

100

Depth (km)

300oC

50

900oC

600oC

1200oC

100

1325oC

Figure by MIT OCW.

Figure 5.10: Elastic thickness.
For the heat flow proper we take z = 0 (q is measured at the surface!) so
that η = 0 and
q = −k

1
(Tm − Ts )
√
⇒q∼ √

πκt
t

(5.17)

with k the conductivity (do not confuse with κ, the diffusivity!). The important result is that according to the half-space cooling model the heat flow
drops of as 1 over the square root of the age of the lithosphere. The heat flow
can be measured, the lithospheric age t determined from, for instance, magnetic
anomalies, and if we assume values for the conductivity and diffusivity, Eq.
(5.17) can be used to determine the temperature difference between the top and
the bottom of the plate
√
πκt
Tm − Ts = q
(5.18)
k
Parsons & Sclater did this (JGR, 1977); assuming k = 3.13 JK−1 m−1 s−1 ,
CP = 1.17×103 Jkg−1 K−1 , and ρ = 3.3×103 kgm−3 and using q = 471 mWm−2
as the best fit to the data they found: Tm − Ts = 1350◦ ± 275◦ C.


5.3. THERMAL STRUCTURE OF THE OCEANIC LITHOSPHERE 205
Comparison to observed heat flow data:

qs (hfu)

6

4

2

0

0

50

100

150

t (106 yr)
Figure by MIT OCW.

Figure 5.11: .
Near the ridge crest the observed heat flow is significantly lower than the
heat flow predicted from the cooling half-space model. In old oceanic basins the
heat flow seems to level off at around 46 mW/m2 , which suggests that beyond
a certain age of the lithosphere the rate of conductive cooling either becomes
smaller or the cooling is partly off set by additional heat production. Possible
sources of heat which could prevent the half-space cooling are:

1. radioactivity (A is not zero!)

2. shear heating

3. small-scale convection below plate

4. hot upwelings (plumes)

206

CHAPTER 5.

GEODYNAMICS

Intermezzo 5.2 Plate-cooling Models
There are two basic models for the description of the cooling oceanic lithosphere,
a cooling of a uniform half space and the cooling of a layer with some finite
thickness. The former is referred to as the half-space model (first described
in this context by Turcotte and Oxburgh, 1967); The latter is also known as the
plate model (first described by McKenzie, 1967).
Both models assume that the plate moves as a unit, that the surface of the
lithosphere is at an isothermal condition of 0◦ C, and that the main method of
heat transfer is conduction (a good assumption, except at the ridge crest). The
major difference (apart from the mathematical description) is that in the halfspace model the base of the lithosphere is defined by an isotherm (for instance
1300◦ C) so that plate thickness can grow indefinitely whereas in the plate model
the plate thickness is limited by some thickness L.
The two models give the same results for young plates near the ridge crest, i.e.
the thickness is such that the ”bottom” of the lithosphere is not yet ”sensed”.
However, they differ significantly after 50 Myr for heat flow predictions and 70
Myr for topography predictions. It was realized early on that at large distances
from the ridge (i.e., large ages of the lithosphere) the oceans were not as deep
and heat flow not as low as expected from the half-space cooling model (there
does not seem to be much thermal difference between lithosphere of 80 and 160
Myr of age). The plate model was proposed to get a better fit to the data, but
its conceptual disadvantage is that it does not explain why the lithosphere has
a maximum thickness of L. The half space model makes more sense physically

and its mathematical description is more straightforward. Therefore, we will
discuss only the half space cooling model, but we will also give some relevant
comparisons with the plate model.

5.4

Thermal structure of the oceanic lithosphere

Bathymetry
The second thermal effect on the evolution of the cooling lithosphere is its
subsidence or the increase in ocean depth with increasing age. This happens
because when the mantle material cools and solidifies after melting at the MOR
it is heavier than the density of the underlying mantle. Since we have seen that
the plate thickens with increasing distance from the MOR and if the plate is
not allowed to subside this would result in the increase in hydrostatic pressure
at some reference depth. In other words the plate would not be in hydrostatic
equilibrium. But when the lithosphere subsides, denser material will be replaced
by lighter water so that the total weight of a certain column remains the same.
The requirement of hydrostatic equilibrium gives us the lateral variation in
depth to the ocean floor. Application of the isostatic principle gives us the
correct ocean floor topography.
Let ρw and ρm represent the density of water (ρw ∼ 1000 kg/m3 ) and the
mantle/asthenosphere (ρm ∼ 3300 kg/m3 ), respectively, and ρ(z, t) the density
as a function of time and depth within the cooling plate. The system is in
hydrostatic equilibrium when the total hydrostatic pressure of a column under
the ridge crest at depth w + zL is the same as the pressure of a column of the


5.4. THERMAL STRUCTURE OF THE OCEANIC LITHOSPHERE 207

x2

x1

x

Ridge

w

�w

WATER
�

yL

LITHOSPHERE

�m
|x |
t2 = u2

|x |
t1 = u1

y

ASTHENOSPHERE

Figure by MIT OCW.

Figure 5.12: Oceanic isostasy.
same width at any distance from the MOR:
�zL

ρ(z, t) dz

(5.19)

�zL
w(ρw − ρm ) + (ρ − ρm ) dz = 0

(5.20)

ρm (w + zL ) = wρw +

0

or

0

According to (5.20) the mass deficiency caused by w(ρw − ρm ) (which is
less than 0!) is balanced by the difference between ρ − ρm (>0!) integrated over
the (as yet unknown) lithospheric thickness. The lithosphere is thus heavier
than the underlying half space! (Assuming, as we do here, that the lithosphere
has the same composition as the asthenosphere). This increase in density is due
to cooling; the relationship between the change in density due to a change in
temperature is given by

dρ = −αρdT ⇒ ρ − ρm

=

−αρ(T − Tm )

⇒ ρ(z, t) = ρm + αρ(Tm − T (z, t))

(5.21)
(5.22)

with α the coefficient of thermal expansion, so that
�zL
w(ρw − ρm ) + ρm α (Tm − T ) dz = 0
0

With T = T (z, t), this gives (verify!!)

(5.23)


208

CHAPTER 5.

GEODYNAMICS

�
��
�zL �

z
dz
w(ρw − ρm ) = ρm α
(Tm − Ts ) − (Tm − Ts ) erf √
2 κt
0

�
��
�zL �
z
= ρm α(Tm − Ts )
1 − erf √
dz
2 κt
0

= ρm α(Tm − Ts )

�zL

erfc

�

0

z
√
2 κt

�

dz
(5.24)

we can change the integration boundary from zL to ∞ because at the base
of the lithosphere T → Tm and ρ → ρm so that we can take the compensation
a t any depth beneath the base of the cooling lithosphere
(and erfc integrated
√
from zL to ∞ is very small). If we also use η = z/(2 κπ) (see above) than we
can write

w(ρw − ρm ) = ρm α(Tm − Ts )

�∞
0

�

z
erfc √
2 κt

�

dz

∞

√ �
= 2ρm α(Tm − Ts ) κt
erfc(η)

(5.25)

0

now use

�∞
0

w(t) =

erfc(q) dq =

√1
π

2ρm α(Tm − Ts )
(ρw − ρm )

to get

�

κt
π

(5.26)

with w(t) the depth below the ridge crest; if the crest is at depth w0 (5.26)
becomes
�
2ρm α(Tm − Ts ) κt
w(t) = w0 +
(5.27)
(ρw − ρm )
π
So from the half-space cooling model it follows that the depth to the sea floor
Sclater
increases as the square root of age! Using α = 3.2×10−5◦ C−1 , Pars ons & √
(1977) found from the fit to bathymetry data gives: w(t) = 2500 + +350 t [m],
for t < 70 Ma.
There is still a lively debate about the details of the parameters that give
the best fit to the model, see, for instance the papers by Stein & Stein (Nature,
1992) and McNutt (Reviews of Geophysics, 1995). But despite the ongoing


5.4. THERMAL STRUCTURE OF THE OCEANIC LITHOSPHERE 209
discussions it is fair to say that these models have been very successful in predicting heat flow, topography, gravity, and have thus played a major role in
the understanding of the evolution of oceanic lithosphere with time. The typical game that is played by such successful theoretical models, in particular ones
that are so simple (= easy + fast to compute) as the cooling models is to predict
the first order behavior of a certain process and take out that trend from the
observed data. In this case, the residual signal is then analyzed for deviations
from the simple conduction model. The addition of heat to the system (for instance by plumes) could cause anomalous topography (”thermal topography”)
whereas the effect of deep dynamic processes in the Earth’s mantle can cause
”dynamic topography”. Removing the effects of conduction alone thus helps to

isolate the structural signal due to other processes. This is likely to continue,
perhaps with the new model (GHD1) by Stein & Stein (1992) instead of that
by earlier workers; since regional differences are often larger than the residual
between observed and predicted heat flow and depth curves one could question
how useful a (set of) simple model(s) is (are). For instance, if one allows the
thermal expansion coefficient as a free parameters in the inversions, one might
also look into allowing lateral variation of this coefficient. Davies and Richards
argue that the success of the cooling models in predicting the topography and
heat flow over almost the entire age range of oceanic lithosphere (they attribute
the deviations to the choice of the wrong sites for data — which is rather questionable) indicates that conductive cooling is the predominant mode of heat loss
of most of the lithosphere (about 85% of the heat lost from the mantle flows
through oceanic lithosphere), which suggests that the lithosphere is the boundary layer of a convective system with a typical scale length defined by the plates
(plate-scale flow). They follow up on a concept tossed up by Brad Hager that
the oceanic lithosphere organizes the flow in the deeper mantle. It is for arguments such as these that question as to whether or not the topography levels
off after, say, 80Ma, in not merely of interest to statisticians. It is quite clear
that the details of the bathymetry of the oceans still contain significant keys to
the understanding of dynamic processes in the deep interior of the Earth. One
of the remarkable aspects of the square root of time variation of ocean depth is
that it does a very good job in describing the true bathymetry, even at distances
pretty close to the MORs. This indicates that conduction alone is likely to be
the predominant mode of heat loss, even close to the MOR. The absence of any
substantial dynamic topography near the ridge crest suggests that the active,
convection related upwellings are not significant. The upwelling is ”passive”: the
plates are pulled apart (mainly as the result of the gravitational force, the slab
pull, acting on the subducting slabs) and the asthenospheric mantle beneath
the ridges flows to shallower depth to fill the vacancy. In doing so the material
will cross the solidus, the temperature at which rock melts (which decreases
with decreasing depth) so that the material melts. This process is known as decompression melting (see Turcotte & Schubert, Chapter 1), which results in
a shallow magma chamber beneath the MOR instead of a very deep plume-like
conduit.

210

CHAPTER 5.

GEODYNAMICS

Temperature (oC)

Depth (km)

1000
0

50

1200

1400

Depth at which partial
melting begins

Temperature of
ascending mantle
rock

Solidus

100

Figure by MIT OCW.

Figure 5.13: Pressure-release melting.

5.5

Bending, or flexure, of thin elastic plate

Introduction
We have seen that upon rifting away from the MOR the lithosphere thickens (the
base of the thermal lithosphere is defined by an isotherm, usually Tm ≈ 1300◦C)
and subsides, and that the cooled lithosphere is more dense than the underlying
mantle. In other words, it forms a gravitationally unstable layer. Why does it
stay atop the asthenosphere instead of sinking down to produce a more stable
density stratification? That is because upon cooling the lithosphere also acquires
strength. Its weight is supported by its strength; the lithosphere can sustain
large stresses before it breaks. The initiation of subduction is therefore less
trivial than one might think and our understanding of this process is still far
from complete.
The strength of the lithosphere has important implications:
1. it means that the lithosphere can support loads, for instance by seamounts
2. the lithosphere, at least the top half of it, is seismogenic
3. lithosphere does not simply sink into the mantle at trenches, but it bends
or flexes, so that it influences the style of deformation along convergent
plate boundaries.
Investigation of the bending or flexure of the plate provides important information about the mechanical properties of the lithospheric plate. We will see
that the nature of the bending is largely dependent on the flexural rigidity,
which in turn depends on the elastic parameters of the lithosphere and on the

elastic thickness of the plate.


211

5.5. BENDING, OR FLEXURE, OF THIN ELASTIC PLATE

An important aspect of the derivations given below is that the thickness
of the elastic lithosphere can often be determined from surprisingly simple observations and without knowledge of the actual load. In addition, we will see
that if the bending of the lithosphere is relatively small the entire mechanical
lithosphere behaves as an elastic plate; if the bending is large some of the deformation takes place by means of ductile creep and the part of the lithosphere
that behaves elastically is thinner than the mechanical lithosphere proper.

5.5.1

Basic theory

To derive the equations for the bending of a thin elastic plate we need to
1. apply laws for equilibrium:
sum�of the forces is zero and the sum of all
�
moments is zero:
F = 0 and
M =0

2. define the constitutive relations between applied stress σ and resultant
strain ǫ
3. assume that the deflection w ≪ L, the typical length scale of the system,
and h, the thickness of the elastic plate ≪ L. The latter criterion (#3) is
to justify the use of linear elasticity.

Figure 5.14: Deflection of a plate under a load.
In a 2D situation, i.e., there is no change in the direction of y, the bending
of a homogeneous, elastic plate due to a load V (x) can be described by the
fourth-order differential equation that is well known in elastic beam theory in
engineering:
D

d4 w
d2 w
+ P 2 = V (x)
4
dx
dx

(5.28)

with w = w(x) the deflection, i.e., the vertical displacement of the plate,
which is, in fact, the ocean depth(!), D the flexural rigidity, and P a horizontal
force.
The flexural rigidity depends on elastic parameters of the plate as well as on
the thickness of the plate:
D=

Eh3
12(1 − ν 2 )

(5.29)

212

CHAPTER 5.

GEODYNAMICS

with E the Young’s modulus and ν the Poisson’s ratio, which depend on the
elastic moduli µ and λ (See Fowler, Appendix 2).
The bending of the plate results in bending (or fiber) stresses within the
plate, σxx ; depending on how the plate is bent, one half of the plate will be in
compression while the other half is in extension. In the center of the plate the
stress goes to zero; this defines the neutral line or plane. If the bending is not
too large, the stress will increase linearly with increasing distance z ′ away from
the neutral line and reaches a maximum at z ′ = ±h/2. The bending stress is
also dependent on the elastic properties of the plate and on how much the plate
is bent; σxx ∼ elastic moduli ×z ′ × curvature, with the curvature defined as the
(negative of the) change in the slope d/dx(dw/dx):
σxx = −

Eh3 ′ d2 w
z
1 − ν 2 dx2

(5.30)

y
αxx

M

comp

h
y=0

x

M

ext
αxx
Figure by MIT OCW.

Figure 5.15: Curvature of an elastic plate.
This stress is important to understand where the plate may break (seismicity!) with normal faulting above and reverse faulting beneath the neutral
line.
The integrated effect of the bending stress is the bending moment M ,
which results in the rotation of the plate, or a plate segment, in the x − z plane.
h

M=

�− 2

σxx z ′ dz ′

(5.31)

h
2

Equation (5.28) is generally applicable to problems involving the bending of
a thin elastic plate. It plays a fundamental role in the study of such problems
as the folding of geologic strata, the development of sedimentary basins, the
post-glacial rebound, the proper modeling of isostasy, and in the understanding


5.5. BENDING, OR FLEXURE, OF THIN ELASTIC PLATE

213

of seismicity. In class we will look at two important cases: (1) loading by sea
mounts, and (2) bending at the trench.
Before we can do this we have to look a bit more carefully at the dynamics of
the system. If we apply bending theory to study lithospheric flexure we have to
realize that if some load V or moment M causes a deflection of the plate there
will be a hydrostatic restoring force owing to the replacement of heavy mantle
material by lighter water or crustal rock. The magnitude of the restoring force
can easily be found by applying the isostasy principle and the effective load is
thus the applied load minus the restoring force (all per unit length in the y
direction): V = Vapplied − Δρwg with w the deflection and g the gravitational
acceleration. This formulation also makes clear that lithospheric flexure is in fact
a compensation mechanism for isostasy! For oceanic lithosphere Δρ = ρm − ρw
and for continental flexure Δρ = ρm − ρc . The bending equation that we will
consider is thus:
D

d4 w
d2 w
+ P 2 + Δρwg = V (x)

4
dx
dx

(5.32)

Loading by sea mounts
Let’s assume a line load in the form of a chain of sea mounts, for example
Hawaii.

Figure 5.16: Deflection of an elastic plate under a line load.
Let V0 be the load applied at x = 0 and V (x) = 0 for x �= 0. With this
approximation we can solve the homogeneous form of (5.32) for x > 0 and take
the mirror image to get the deflection w(x) for x < 0. If we also ignore the
horizontal applied force P we have to solve
D

d4 w
+ Δρwg = 0
dx4

(5.33)

The general solution of (5.33) is
x

w(x) = e α

�
�

x
x
x�
x
x�
A cos + B sin
+ e− α C cos + D sin
α
α
α
α

(5.34)


214

CHAPTER 5.

GEODYNAMICS

with α the flexural parameter, which plays a central role in the extraction
of structural information from the observed data:
α=

�

4D
Δρg

� 14

(5.35)

Courtesy of Annual Reviews Inc. Used with permission.

Figure 5.17: .
The constants A − D can be determined from the boundary conditions. In
this case we can apply the general requirement that w(x) → 0 for x → ∞ so that
A = B = 0, and we also require that the plate be horizontal directly beneath
x = 0: dw/dx = 0 for x = 0 so that C = D: the solution becomes
x

w(x) = Ce− α

�

cos

x
x�
+ sin
α
α

(5.36)

From this we can now begin to see the power of this method. The deflection
w as a function of distance is an oscillation with period x/α and with an exponentially decaying amplitude. This indicates that we can determine α directly
from observed bathymetry profiles w(x), and from equations (5.36) and (5.29)

we can determine the elastic thickness h under the assumption of values for the
elastic parameters (Young’s modulus and Poisson’s ratio). The flexural parameter α has a dimension of distance, and defines, in fact, a typical length scale of
the deflection (as a function of the “strength” of the plate).
The constant C can be determined from the deflection at x = 0 and it can
be shown (Turcotte & Schubert) that C = (V0 α3 )/(8D) ≡ w0 , the deflection


5.5. BENDING, OR FLEXURE, OF THIN ELASTIC PLATE

215

x/�

0

1

2

3

w/w0

x0/�

4

5

xb/�

0.5

1

Figure by MIT OCW.

Figure 5.18: A deflection profile.
beneath the center of the load. The final expression for the deflection due to a
line load is then
V0 α3 − x �
x
x�
x≥0
(5.37)
e α cos + sin
w(x) =
8D
α
α
Let’s now look at a few properties of the solution:
• The half-width of the depression can be found by solving for w = 0. From
(5.37) it follows that cos(x0 /α) = − sin(x0 /α) or x0 /α = tan−1 (−1) ⇒
x0 = α(3π/4 + nπ), n = 0, 1, 2, 3 . . . For n = 0 the half-width of the
depression is found to be α3π/4.
• The height, wb , and location, xb , of forebulge ⇒ find the optima of the
solution (5.37). By solving dw/dx = 0 we find that sin(x/α) must be zero
⇒ x = nπα, and for those optima w = w0 e−nπ , n = 0, 1, 2, 3 . . . For the
location of the forebulge: n = 1, xb = πα and the height of the forebulge
wb = −w0 e−π or wb = −0.04w0 (very small!).